In a class of $8$, there are $2$ students who are secretly robots. If the teacher chooses $4$ students, what is the probability that none of the four of them are secretly robots?
Answer: We can think about this problem as the probability of $4$ events happening. The first event is the teacher choosing one student who is not secretly a robot. The second event is the teacher choosing another student who is not secretly a robot, given that the teacher already chose someone who is not secretly a robot, and so on. The probabilty that the teacher will choose someone who is not secretly a robot is the number of students who are not secretly robots divided by the total number of students: $\dfrac{6} {8}$ Once the teacher's chosen one student, there are only $7$ left. There's also one fewer student who is not secretly a robot, since the teacher isn't going to pick the same student twice. So, the probability that the teacher picks a second student who also is not secretly a robot is $\dfrac{5} {7}$ The probability of the teacher picking two students who are not secretly robots must then be $\dfrac{6} {8} \cdot \dfrac{5} {7}$ We can continue using the same logic for the rest of the students the teacher picks. So, the probability of the teacher picking $4$ students such that none of them are secretly robots is $\dfrac{6}{8}\cdot\dfrac{5}{7}\cdot\dfrac{4}{6}\cdot\dfrac{3}{5} = \dfrac{3}{14}$